Basic Mathematical Concepts Used in Wireless Networks
There are few basic mathematical concepts that are essential to the understanding of the techniques used in wireless networks, this chapter covers most of them. It is very helpful to understand how these concepts evolved historically and what they tried to achieve. For this reason, we give a historical background of each concept, keeping the mathematics to a minimum, and, instead, emphasizing the principles involved and their outcome.
22.1 Circle Relationships
A circle is a very important geometric figure used to represent many relationships. It is used, for example, to represent locations on the Earth's globe (coordinates) and phase and amplitude relationships.
The calculation of the circle length, or circle linearization, was not an easy task and efforts have been made since antiquity to express it as a function of the circle's radius. It was soon established that the ratio between the length and the radius was a constant, but the exact value was unknown and it could only be approximated by use of fractions.
Another challenge was to calculate the area encompassed by the circle, or circle quadrature. Again it was suspected that there was a constant relationship between the area and the radius, similar to the one used in the circle linearization, but that constant could not be calculated either.
It was Archimedes (287–212bc) who demonstrated that the same constant applied to both relationships and he called it π, ...